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Abstract

For the benefit of designing scalable, fault resistant optical neural networks (ONNs), we investigate the effects architectural designs have on the ONNs’ robustness to imprecise components. We train two ONNs – one with a more tunable design (GridNet) and one with better fault tolerance (FFTNet) – to classify handwritten digits. When simulated without any imperfections, GridNet yields a better accuracy (∼98%) than FFTNet (∼95%). However, under a small amount of error in their photonic components, the more fault tolerant FFTNet overtakes GridNet. We further provide thorough quantitative and qualitative analyses of ONNs’ sensitivity to varying levels and types of imprecisions. Our results offer guidelines for the principled design of fault-tolerant ONNs as well as a foundation for further research.

Figures (16)

Fig. 1 a) A schematic of a universal 8×4 optical linear multiplier with two unitary multipliers (red) consisting of MZIs in a grid-like layout and a diagonal layer (yellow). The MZIs of GridUnitary multipliers are indexed according to their layer depth (l) and dimension (d). Symbols at the top represent the mathematical operations performed by the various modules. Inset: A MZI with two 50:50 beamsplitters and two tunable phaseshifters b) An FFT-like, non-universal multiplier with FFTUnitary multipliers (blue).

Fig. 2 Network design used for the MNIST classification task. GridNet used universal unitary multipliers while FFTNet used FFT-Unitary multipliers. See Fig. 1 for details of physical implementation of the three linear layers.

Fig. 3 Visualizing the degradation of ONN outputs, FFTNet is seen to be much more robust than GridNet. Identical input is fed through GridNet (a, b) and FFTNet (c, d), simulated with ideal components (a, c) and imprecise components (b, d) with σBS = 0.01 and σPS = 0.01 rad. Imprecise networks are simulated 100 times and their mean output is represented by bar plots. Error bars represent the 20th to 80th percentile range.

Fig. 4 The decrease in classification accuracy is visualized for GridNet and FFTNet. (a,b) The two networks were tested with simulated noise of various levels for 20 runs. The mean accuracy is plotted as a function of σPS and σBS. Note the difference in color map ranges between the two plots. (c) The accuracies of GridNet and FFTNet are compared along the σPS = σBS cutline.

Fig. 5 The architecture of a) StackedFFT and b) TruncGrid shown with FFTUnitary and GridUnitary from which they were derived. For clarity, the dimension, here, is N = 24 = 16 so FFTUnitary was stacked four times and GridUnitary was truncated at the fourth layer. In the experiments described in this section, the dimension was taken to be N = 28 = 256.

Fig. 6 With the same layer depth, multipliers with FFT-like architectures are shown to be more robust. The fidelity between the error-free and imprecise transfer matrices is plotted as a function of increasing error. Two sets of comparisons between unitary multipliers of the same depth are made. a) Both StackedFFT and GridUnitary have N = 256 layers of MZIs. b) TruncGrid and FFTNet have log N = 8 layers.

Fig. 7 Change in accuracy due to localized imprecision in layer 2 of GridNet with randomized singular values. A large amount of imprecision (σPS = 0.1 rad) is introduced to 8×8 blocks of MZIs in an otherwise error-free GridNet. The resulting change in accuracy of the network is plotted as a function of the position of the MZI block in GridUnitary multipliers V2† and U2 (coordinates defined as in Fig. 1(a)). The transmissivity of each waveguide through the diagonal layer Σ2 is also plotted (center panel).

Fig. 8 Effects of localized imprecision in layer 2 of GridNet with ordered singular values. Similar to Fig. 7, except GridNet has its singular values ordered. Therefore, the transmissivity is also ordered (center panel).

Fig. 11 The degradation of ONN outputs visualized through confusion matrices. Each confusion matrix shows how often each target class (row) is predicted as each of the ten possible classs (column). Both networks, GridNet (a, b, c) and FFTNet (d, e, f) are evaluated. First in the ideal case (a, d) then, with increasing errors (b, e and c ,f). Note the logarithmic scaling.

Fig. 12 The effects of quantization is shown for both GridNet and FFTNet. 10 instances of GridNet (blue) and FFTNet (red) were trained then quantized to varying levels. The mean classification accuracy at each level is shown by bar plots. The 20-80%quantiles are shown with error bars. The dotted horizontal line denotes the full precision accuracy.

Fig. 13 The central MZIs of GridNet has lower variance in internal phase shifts (θ). a) The spatial distribution of internal phase shift (θd,l) of MZIs in U2 of GridNet. Reference Fig. 1(a) for coordinates and Fig. 2 for location of U2 in context of network architecture. b) Histogram of phase shifts near the center (red), edge (green), and corner (blue) of the GridUnitary multiplier. These phases are obtained from multiple instances of trained GridNets with random initialization.

Fig. 14 The variance of internal phase shifts of FFTNet is uniform spatially (a) Spatial distribution of phase shifts for a FFTUnitary multiplier. The MZIs are ordered as shown in Fig. 1(b). (b) Histogram of phase shifts of FFTUnitary near the center (red) and top (green). These phases are obtained from mulitple trained FFTNets with random initialization.

Fig. 15 a) A schematic of BlockFFTUnitary. Blocks of MZIs in dashed, blue boxes are similar to GridNet. The crossing waveguide, similar to those in FFTNet are between the blocks. b) The distribution of phases after being trained. The dashed white lines denote the locations of the crossing waveguides.